Subgroups — #13118359

Order 1664.

500 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88200	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718648	C₂	1	1	yes	maximal, characteristic
5	#11718646	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#6103	C₂	1	1	yes	maximal, characteristic
7	#11718596	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718338	C₂	1	2	yes	maximal
9	#11718331	C₂	1	2	yes	maximal
10	#11718322	C₂	1	2	yes	maximal, complemented
11	#11718314	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	#88200	—	2	4	no	—
17	Q₈×Dic₁₃	—	2	4	no	—
18	C₂×#6103	C₂²	1	2	yes	—
19	C₂×#6103	C₂²	1	1	yes	characteristic
20	C₂×#6103	C₂²	1	2	yes	—
21	C₂×#6103	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
28	C₂×#6101	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
30	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
31	C₁₃×#95	C₂²	1	2	yes	—
32	C₁₃×#95	C₂²	1	2	yes	—
33	#88071	C₂²	1	1	yes	characteristic
34	#88071	C₂²	1	2	yes	—
35	#88071	C₂²	1	2	yes	—
36	#88071	C₂²	1	1	yes	characteristic
37	C₂³×D₁₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₁₃	—	2	2	no	—
39	C₂×Q₈×C₁₃	—	2	2	no	—
40	C₂×Q₈×C₁₃	D₄	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂³	1	2	yes	—
48	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₃	—	2	2	no	—
52	C₂×C₄×D₁₃	—	2	2	no	—
53	C₂×C₄×D₁₃	—	2	2	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₄²×C₁₃	—	2	2	no	—
56	#6103	—	2	4	no	—
57	#6103	—	2	2	no	—
58	#6103	—	4	2	no	—
59	#6103	Q₈	1	4	yes	complemented
60	C₄⋊Dic₁₃	—	2	4	no	—
61	C₄⋊Dic₁₃	—	2	2	no	—
62	C₄⋊Dic₁₃	—	2	2	no	—
63	C₄⋊Dic₁₃	—	2	4	no	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	#6101	—	2	4	no	—
67	C₄×Dic₁₃	—	2	4	no	—
68	C₄×Dic₁₃	—	2	2	no	—
69	#2423	—	13	1	no	maximal, Sylow
70	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂²×D₁₃	—	2	1	no	—
72	C₂²×D₁₃	—	2	1	no	—
73	C₂²×D₁₃	—	2	1	no	—
74	C₂²×D₁₃	—	2	1	no	—
75	C₂²×D₁₃	—	2	1	no	—
76	C₂²×D₁₃	—	2	1	no	—
77	C₂²×D₁₃	C₂×Q₈	1	2	yes	—
78	Q₈×C₁₃	—	2	8	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
89	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₁₃	#41	1	2	yes	—
92	C₂×C₄×C₁₃	#41	1	2	yes	—
93	C₂×C₄×C₁₃	#41	1	2	yes	—
94	C₂×Dic₁₃	—	2	1	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	—	2	1	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	—	2	2	no	—
102	C₂×Dic₁₃	—	2	1	no	—
103	C₂×Dic₁₃	—	2	1	no	—
104	C₂×Dic₁₃	—	2	1	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
109	C₄×D₁₃	—	4	4	no	—
110	C₂×C₂²⋊Q₈	—	13	1	no	—
111	C₂×C₄×Q₈	—	13	1	no	—
112	#399	—	13	2	no	—
113	#398	—	13	2	no	—
114	#391	D₁₃	1	1	yes	characteristic, complemented
115	#389	—	13	1	no	—
116	#388	—	13	2	no	—
117	#387	—	13	2	no	—
118	#384	—	13	1	no	—
119	#380	—	13	1	no	—
120	C₄×C₂²⋊C₄	—	13	1	no	—
121	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×D₁₃	—	2	2	no	—
129	C₂×D₁₃	—	2	2	no	—
130	C₂×D₁₃	—	2	2	no	—
131	C₂×D₁₃	—	4	1	no	—
132	C₂×D₁₃	—	4	1	no	—
133	C₂×D₁₃	—	4	1	no	—
134	C₂×D₁₃	—	4	1	no	—
135	C₄×C₁₃	—	2	4	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	4	2	no	—
139	C₄×C₁₃	#123	1	4	yes	—
140	Dic₁₃	—	2	2	no	—
141	Dic₁₃	—	4	1	no	—
142	Dic₁₃	—	4	2	no	—
143	Dic₁₃	—	4	1	no	—
144	Dic₁₃	—	4	2	no	—
145	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
146	C₂³×C₄	—	13	1	no	—
147	C₂²⋊Q₈	—	26	4	no	—
148	C₄×Q₈	—	26	4	no	—
149	C₂×C₄⋊C₄	—	13	1	no	—
150	C₂×C₄⋊C₄	—	13	2	no	—
151	C₂×C₄⋊C₄	—	13	2	no	—
152	C₂×C₄⋊C₄	—	13	1	no	—
153	C₂×C₄⋊C₄	—	13	2	no	—
154	C₂×C₄⋊C₄	—	13	2	no	—
155	C₂×C₄⋊C₄	—	13	2	no	—
156	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	—	13	1	no	—
158	C₂×C₂²⋊C₄	—	13	2	no	—
159	C₂×C₂²⋊C₄	—	13	1	no	—
160	C₂×C₂²⋊C₄	—	13	2	no	—
161	C₂×C₄²	—	13	1	no	—
162	C₂×C₄²	—	13	2	no	—
163	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
164	#95	—	13	1	no	—
165	#95	—	13	1	no	—
166	#95	—	13	2	no	—
167	#95	—	13	2	no	—
168	#95	C₂×D₁₃	1	2	yes	—
169	#95	C₂×D₁₃	1	2	yes	—
170	C₂×C₁₃	C₂×#123	1	1	yes	characteristic
171	C₂×C₁₃	#533	1	1	yes	characteristic
172	C₂×C₁₃	#542	1	1	yes	characteristic
173	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
174	C₂×C₁₃	#550	1	1	yes	characteristic
175	C₂×C₁₃	#552	1	1	yes	characteristic
176	C₂×C₁₃	#554	1	1	yes	characteristic
177	D₁₃	—	4	2	no	—
178	C₂⁴	—	13	1	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	D₅₂	1	4	yes	—
182	C₂²×C₄	—	13	1	no	—
183	C₂²×C₄	—	13	1	no	—
184	C₂²×C₄	—	13	1	no	—
185	C₂²×C₄	—	13	2	no	—
186	C₂²×C₄	—	13	2	no	—
187	C₂²×C₄	—	26	2	no	—
188	C₂²×C₄	—	26	2	no	—
189	C₂²×C₄	—	26	2	no	—
190	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
191	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	26	4	no	—
197	C₄⋊C₄	—	26	2	no	—
198	C₄⋊C₄	—	26	4	no	—
199	C₄⋊C₄	—	26	2	no	—
200	C₄⋊C₄	—	26	4	no	—
201	C₄⋊C₄	—	52	2	no	—
202	C₄⋊C₄	—	52	2	no	—
203	C₂²⋊C₄	—	13	4	no	—
204	C₂²⋊C₄	—	26	4	no	—
205	C₂²⋊C₄	—	26	2	no	—
206	C₂²⋊C₄	—	52	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	26	4	no	—
209	C₄²	—	26	2	no	—
210	C₁₃	#2423	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	13	2	no	—
212	C₂³	—	26	1	no	—
213	C₂³	—	26	1	no	—
214	C₂³	—	26	1	no	—
215	C₂³	—	26	1	no	—
216	C₂³	—	26	1	no	—
217	C₂³	—	26	1	no	—
218	C₂³	C₂³×D₁₃	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	13	2	no	—
230	C₂×C₄	—	26	1	no	—
231	C₂×C₄	—	26	1	no	—
232	C₂×C₄	—	26	1	no	—
233	C₂×C₄	—	26	2	no	—
234	C₂×C₄	—	26	2	no	—
235	C₂×C₄	—	26	1	no	—
236	C₂×C₄	—	26	1	no	—
237	C₂×C₄	—	26	1	no	—
238	C₂×C₄	—	26	1	no	—
239	C₂×C₄	—	26	2	no	—
240	C₂×C₄	—	26	2	no	—
241	C₂×C₄	—	26	2	no	—
242	C₂×C₄	—	26	1	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	—	52	4	no	—
245	C₂×C₄	C₂×D₅₂	1	4	yes	—
246	C₂×C₄	C₂×D₅₂	1	2	yes	—
247	C₂×C₄	#6129	1	2	yes	—
248	C₂×C₄	#6129	1	2	yes	—
249	C₂×C₄	Q₈×D₁₃	1	2	yes	—
250	C₂×C₄	#6131	1	2	yes	—
251	C₂²	—	26	2	no	—
252	C₂²	—	26	2	no	—
253	C₂²	—	26	2	no	—
254	C₂²	—	52	1	no	—
255	C₂²	—	52	1	no	—
256	C₂²	—	52	1	no	—
257	C₂²	—	52	1	no	—
258	C₂²	C₂²×D₅₂	1	1	yes	characteristic
259	C₂²	C₂×#6129	1	1	yes	characteristic
260	C₂²	C₂×#6129	1	1	yes	characteristic
261	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
262	C₂²	C₂×#6131	1	1	yes	characteristic
263	C₂²	#88256	1	1	yes	characteristic
264	C₂²	#88257	1	1	yes	characteristic
265	C₄	—	2	4	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	26	2	no	—
270	C₄	—	52	1	no	—
271	C₄	—	52	1	no	—
272	C₄	—	52	2	no	—
273	C₄	—	52	2	no	—
274	C₄	#88140	1	4	yes	—
275	C₂	—	52	2	no	—
276	C₂	C₂×#88140	1	1	yes	characteristic
277	C₂	Q₈×D₅₂	1	1	yes	characteristic
278	C₂	#11719282	1	1	yes	characteristic
279	C₂	#11719340	1	1	yes	characteristic
280	C₂	#11719347	1	1	yes	characteristic
281	C₂	#11719387	1	1	yes	characteristic
282	C₂	#11719433	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#391	C₂
prime radical	C₂³×D₁₃	C₂³
hypercenter	#391	D₁₃
hypocenter	C₁₃	#2423
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #391

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×Q₈×Dic₁₃	C₂	semidirect
3. C₁₃×#391	C₂	semidirect
5. #11718646	C₂	semidirect
7. #11718596	C₂	semidirect
10. #11718322	C₂	semidirect
11. #11718314	C₂	semidirect
59. #6103	Q₈	semidirect
114. #391	D₁₃	semidirect
210. C₁₃	#2423	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#88200	2
C₂×Q₈×Dic₁₃	2
C₁₃×#391	2
#11718648	2
#11718646	2
C₄×#6103	2
#11718596	2
#11718338	2
#11718331	2
#11718322	2
#11718314	2
#2423	13